Hydrostatic Paradox

Consider several interconnected vessels of very different shapes — some narrow, some wide, some tapering — all joined together at their base and open at the top. When a liquid is poured into any one of them, it is found that the liquid rises to exactly the same level in every one of the vessels, no matter how different their shapes or widths are. This observation seems puzzling at first: one might expect the liquid pressure at the base of a wider vessel to be greater than at the base of a narrower one, which would seem to make the liquid rise higher in the narrower vessel until the extra pressure from vessel C forced liquid up into a narrower vessel B — yet this is never actually observed. Before the principles of hydrostatics were fully understood, this was called the hydrostatic paradox.

The resolution comes directly from Eq. (2.2), $p = h\rho g$: the pressure at a point inside a liquid depends only on the height h of the liquid column directly above that point — it does not depend at all on the shape of the vessel. Since the height of liquid is the same in every one of the interconnected vessels (they all show the same level), the pressure at the base is also the same in every vessel, and the whole connected system is in equilibrium — the liquid in a wide vessel like C simply does not rise up into a narrower vessel like B, because there is no pressure imbalance driving it to. …